Dirac-Fock Wave Functions Research Articles

Relativistic atomic structure calculations of Li-like ions used for plasma diagnostic studies

We have carried out atomic structure calculations using systematically enlarged multiconfiguration Dirac-Fock wavefunctions of Li-like ions of the most prominent plasma impurities (Ar, Ti, Fe, Ni, Kr and W) found in presently working tokamaks. Relativistic Breit interaction and quantum electrodynamic (QED) corrections such as vacuum polarization and self-energy corrections are also included in the calculations prior to the evaluation of low lying energy levels, transition probabilities, oscillator strengths and line strengths. Selective radiative data for electric dipole and magnetic quadrupole transitions are also reported. Special emphasis is given in the computations of fundamental quantities such as oscillator strengths as they are widely used in atomic data and analysis structure (ADAS) databases to evaluate quantities such as effective collision strengths. Present computed values are compared with existing available results on NIST database and few similar earlier computations and a good agreement has been found. We believe that the detailed atomic data with the relativistic and QED corrections will assist in spectroscopic studies such as accurate line identification and plasma modelling work in tokamak plasma, laser induced breakdown spectroscopy (LIBS), highly charged ions clocks and astrophysical observations.

Fine structure resolved excitation cross sections of singly ionized Ga for the modeling and diagnostics of Ga plasmas

Calculation of electron impact excitation cross sections for singly charged Ga ions plays a crucial role in plasma modeling, facilitating the comprehension of plasma behavior, characteristics, and dynamics in diverse domains, such as astrophysics, fusion research, the semiconductor industry, etc. In the available literature, there is a notable scarcity of, or even a complete absence of, these cross sections. Hence, in the present work, electron impact excitation cross sections are calculated for the transitions from the fine structure resolved energy levels of the configurations 4s2 and 4s4p to the fine structure resolved energy levels of the configurations 4s4p , 4s5s , 4p2 and 4s4d of the singly charged Ga ion (Ga+) using the relativistic distorted wave approximation theory with the target states represented by multi configurational Dirac Fock wavefunctions. The cross sections are calculated for projectile electron energy varying from threshold to 500 eV. Furthermore, the electron impact excitation rate coefficients for all the transitions under investigation are also calculated for electron temperatures ranging from 0.5 to 5 eV. In addition, analytic fitting of the rate coefficients is also performed, providing a practical resource for directly utilizing in plasma modeling applications.

Fully relativistic distorted wave calculations of electron impact excitation of gallium atom: Cross sections relevant for plasma kinetic modeling

Electron impact excitation cross sections of neutral Ga are calculated using the relativistic distorted wave approximation with the aid of multiconfigurational Dirac-Fock wavefunctions. The cross sections are calculated from the fine structure resolved levels of the 4s24p manifold, i.e., from the ground 2P1/2 and the first excited state 2P3/2, to the fine structure resolved levels of the manifolds 4s24pdf, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25spdf, 4s26spdf, 4s27spdf, and 4s28spdf for incident electron energies ranging from threshold to 500 eV. Where available, a comparison of the calculated oscillator strengths, transition probabilities, and cross sections with the previously reported results has been performed. The cross sections for the excitation from 4s24p2P1/2 to 4s24f, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25f, 4s26f, 4s27df, and 4s28spdf and all the excitation cross sections from 4s24p2P3/2 are calculated and reported for the first time for a wide range of electron energies. Furthermore, the rate coefficients are also calculated using the present cross sections for an electron temperature varying from 0.5 to 5 eV. An analytic fitting of the calculated rate coefficients is provided to facilitate their inclusion in various plasma kinetic models.

Electron impact excitation cross section calculations of the fine structure transitions of Mo and their applications in the diagnostics of the laser induced Mo plasma

The current work reports a detailed calculation of electron impact excitation cross sections for the fine structure transitions of Mo from the manifolds 4d 55s, 4d 45s 2, 4d 6 and 4d 55p to the manifolds 4d 55s, 4d 45s 2, 4d 6, 4d 55p, 4d 45s5p, 4d 55d, 4d 56s, 4d 45s6s, and 4d 57s, using the relativistic distorted wave approximation for the applications in plasma modelling. Multi-configurational Dirac–Fock wave functions are used in the calculations. The oscillator strength and cross section results are compared with the previous calculations and measurements. A comprehensive collisional radiative (CR) model is developed and used to characterize laser induced molybdenum plasma to ensure that the calculated cross sections can be used for various plasma modelling applications. The current CR model has taken into account the electron impact excitation and de-excitation processes using the calculated consistent cross sections. The electron induced processes are dominant kinetic processes in the laser induced plasma. Furthermore, the diagnostics of the laser induced Mo plasma is done by coupling the current CR model with the experimental laser induced breakdown spectroscopic measurements of Mal et al (2021 Appl. Phys. B 127 52). The plasma parameter, i.e., electron temperature has been calculated using nine measured intensities of the emission lines of Mo, with wavelengths 406.9, 423.3, 438.2, 453.7, 476.0, 550.7, 553.3, 557.0 and 592.9 nm. The results are also compared with the values reported in the Boltzmann plot at various delay times ranging from 0.5 to 5.0 μs.

Study of electron impact elastic scattering from Kr@C60 and Xe@C60 using a fully relativistic approach

In the present work, a detailed study has been reported on electron impact elastic scattering from krypton (Kr) and xenon (Xe) atoms when confined in two different types of C60 potentials viz (a) hard annular square well (ASW) and (b) diffused Gaussian annular square well (GASW). The Dirac equations are solved using these potentials for encaged Kr and Xe in C60. First, bound state Dirac–Fock wave functions of these encaged Kr and Xe atoms are found by utilizing modified general relativistic atomic structure package and thereafter, the charge densities and static potentials of the endohedral Kr@C60 and Xe@C60 are obtained. Further, using these, the Dirac equations are solved by the relativistic partial wave phase shift analysis method and the scattering amplitudes in terms of phase shifts are obtained. Thereafter, the electron elastic differential and integrated cross sections of Kr@C60 and Xe@C60 along with the C60 are calculated in the range of 0.1–15 eV incident electron energies. Presently, no experimental and theoretical results are available to compare our electron scattering cross section results from Kr@C60 and Xe@C60; thus, we have shown the cross section results obtained from ASW and GASW potential and compared them.

K X‐ray satellite structures of bromine and molybdenum

The energies and electric dipole rates of Kα X‐ray satellites due to 1s‐2p transitions in bromine and molybdenum with one to five vacancies in the 2p shell and other shells retaining their normal occupation numbers are computed using Multi‐Configuration Dirac Fock wavefunctions with the inclusion of relativistic and quantum electrodynamic corrections. As fully relativistic calculations on all the Kα X‐ray fine structure satellites with multiple spectator vacancies in the 2p shell of elements under consideration, to the best of our knowledge, are reported in this work for the first time, the accuracy of the E1 data is analyzed in terms of the agreement between the rates in the length and velocity forms. While the intensities of the strong electric dipole lines in the various groups of 1s ‐ 2p transitions obey the well established Z dependence, many of the weak transitions vary randomly with Z and an attempt is made in this work to understand the behaviour of the spin‐forbidden and weak electric dipole lines in the KαL n X‐ray satellite structure of Br, Mo, I, and Au. As the number of closely spaced fine structure transitions range from a few to many depending on the spectator vacancies in the 2p shell, the intensity weighed statistical average satellite energies, line strengths and photo‐ionization cross sections are reported to guide in the reliable interpretation of experimental measurements.

Energies and emission strengths of Kα X-rays from iodine and gold with multiply ionized L shell

The energies and rates of Kα X-ray satellites due to 1s − 2p transitions in iodine and gold with one to multiple vacancies in the L shell are computed using Multi-Configuration Dirac Fock wavefunctions with the inclusion of relativistic and quantum electrodynamics corrections. The calculations have been carried out for I1+ to I6+ with spectator vacancies in the 2p shell and Au1+ to Au8+ with filled and empty 2s shell along with vacancies in the 2p shell. As existing data on the X-ray emission from low ionized iodine and gold are confined to single inner shell vacancy, direct comparison of the present results on the various radiative transitions from iodine and gold with a single vacancy in the K shell, variously ionized L shell and no vacancy in any other shells is possible only for the two well known Kα 1 and Kα 2 diagram lines. Hence to substantiate the reliability of the current results through one to one comparison with existing data on the 1s − 2p transitions, calculations have also been performed on the radiative decay of H-, He-, Be-, and F-like gold with singly ionized 1s , closed 2s , multiply ionized 2p and rest of the shells open for which semi-relativistic data exist. In addition to providing additional check on the reliability of the computed results, this comparison also tries to analyze the impact of higher order relativistic corrections on the K X-ray energies and rates. The intrinsic variations of transition properties with degree of ionization have been analyzed.

Revised and extended calculations with relativistic and QED corrections of selected He-like 3d-elemental ions having astrophysical significance

We have systematically enlarged Multiconfiguration Dirac-Fock wavefunctions for the relativistic configuration interaction calculation to include the Breit interaction and quantum electrodynamic corrections for evaluating fine structure data of He-like ions having non-zero nuclear spin for five 3d group elements Scandium, Vanadium, Manganese, Cobalt and Copper. Computed energy levels, transition probabilities, oscillator strengths and x-ray wavelengths are compared with the available experimental values and theoretical computations. Special emphasis is given on the forbidden transition rates and hyperfine structure calculations as they are of potential interest in astrophysics. An attempt is made to study the individual contributions of the relativistic effects through Breit interaction and the quantum electrodynamic corrections such as vacuum polarization and self-energy correction to the line strengths useful for plasma diagnostic studies relevant to astrophysics. • Atomic structure calculation is done using Multiconfiguration Dirac-Fock method. • Relativistic configuration interaction calculation is used to report atomic data. • Impacts of relativistic and QED corrections on line strengths are discussed. • Hyperfine coupling constants A J & B J are evaluated for plasma diagnostic studies. • Computed atomic data for He-like odd Z 3d ions are compared with available data.

Diagnostics of low-temperature neon plasma through a fine-structure resolved collisional–radiative model

A collisional–radiative (CR) model for low-temperature Ne plasma is developed. Various radiative and collisional processes involving the ground 2p6 and the excited 2p53s, 2p53p, 2p53d, 2p54s and 2p54p states are considered in the plasma. First, we calculate a complete set of required electron-impact excitation cross-sections of Ne, which is an important process in such low-temperature plasma. We used the relativistic distorted wave (RDW) theory and calculated electron excitation cross-sections for the transitions from the ground 2p6 state to the excited 2p53s, 2p53p, 2p53d, 2p54s and 2p54p states, as well as from the excited state 2p53s to the 2p53p and 2p54p excited states of the Ne atom in a wide range of incident electron energies from the threshold to 500 eV. The ground and different excited states of the Ne are represented through the multiconfiguration Dirac–Fock wave functions, which are obtained using the GRASP2K code. To ascertain the reliability of the obtained wave functions, we calculated the oscillator strengths for different dipole allowed transitions, and compared them with the available experimental and theoretical results. Further, the calculated detailed RDW cross-sections are presented and compared with the available experimental and other theoretically calculated values. The cross-sections for 2p53s to 2p54p and 2p53s (J = 1 only) to 2p53p are reported for the first time. The complete set of calculated electron excitation cross-sections of different transitions of Ne along with other processes are used to develop the CR model. The model has been applied to the diagnostics of low-temperature Ne plasma by coupling it to the optical emission and absorption measurements of Boffard et al (2012 J. Phys. D: Appl. Phys. 45 382001, and 2009 Plasma Sources Sci. Technol. 18 035017). The extracted values of electron density, electron temperature and the calculated 1si level populations have been compared with the corresponding measurements available in the pressure range of 5–25 mTorr and are found to be in excellent agreement.

Radiative de-excitations of the 1s2s3s configuration of Li-like ions

The fully relativistic configuration interaction method is used to study the one-electron one-photon (OEOP) and two-electron one-photon (TEOP) transitions between the 1s2s3s and the respective 1s2s2p and 1s2np (n = 2–4 ) groups of Li-like ions with 12 ≤ Z ≤ 54. The transition properties are calculated using multi-configuration Dirac–Fock wavefunctions in the active space approximation taking into account the correlated state functions that have non-negligible contributions to the mixing coefficients. The drastic variations in the level lifetimes due especially to 1s2s3s-1s2np (n = 2, 3) transitions are analyzed in detail. The OEOP transition energies are far less than those of TEOP transitions and have been taken up in this work mainly to investigate the most probable radiative decay channel and also to understand the interplay between electron–electron correlation and the low-frequency limit transverse photon interaction on the mixing coefficients of J = 3/2 states of 1s2s2p around Z = 21. The present results are compared with other available data.

Study of energies and radiative properties of He-like ions within a dense plasma

On the basis of the tensor expression for the Breit-Pauli Hamiltonian and the irreducible tensor theory for many-electron atoms, we have proposed a general analytical method for calculating atomic structures of highly charged ions under a dense plasma environment. In this method, the standard ion-sphere (IS) potential is used to represent the interaction between the charge particles. The Racah wave functions are obtained as the linear combinations of the multi-Slater wave functions. The angular interactions and spin sums are worked out using an irreducible theory of complex systems. Analytical formulas for the calculation of relativistic corrections, such as the mass correction, the one-body Darwin correction, the two-body Darwin correction, and the spin-spin contact interaction, are derived. Energy levels and radiative properties of He-like Al11+ and Ni26+ ions are considered for illustrative purposes. Systematic trends with respect to both the nuclear charge Z and the electron density Ne are observed for all calculated quantities. Self-consistent calculations of dense plasma effects on the atomic structure, based on the same IS potential, are also presented using fully relativistic multiconfiguration Dirac-Fock (MCDF) wave functions in the active space approximation with the inclusion of a finite nuclear size, the Breit interaction, self-energy, and vacuum polarization for comparison purposes. Comparison and analysis are made between our two sets of results and the available results in the literature. The effects of relativistic correlations are investigated in detail for the first time and are found to play an important role in the stability of these systems. The results of this study should be of great help in the modeling and diagnostics of astrophysical and laboratory plasmas.

Fluorescence yields for doubly excited 2s3s configuration in He-like ions

The wavelengths and rates of single- and double-electron transitions leading to the emission of a single photon between 2s3s and nsn′p (n = 1, 2 and n′ = 2–4) groups in He-like ions with empty K shell have been studied for 15 ions in the range 13 ≤ Z ≤ 54 using fully relativistic multi-configuration Dirac–Fock wavefunctions in the active space approximation. The complete relaxation of atomic orbitals and most important configuration interaction effects have been taken into account. The variations in the admixture of the correlation functions coupled with Breit interaction responsible for the sudden changes in the line intensities of certain transitions from 2s3s have been discussed in detail. The radiative rates of the electric dipole transitions have been evaluated as accurately as possible by taking into account all the non-negligible contributions to the X-ray rates. The Auger rates have been calculated using a scaling procedure on the already existing relativistic radial matrix elements for an initial K hole atom. The present results are compared with other available theoretical data. The computed data can be conveniently used in the analysis of experimental spectra and in the measurements of X-ray fluorescence yields for the doubly excited states under study.

Theoretical energies and radiative transition parameters of W L

Using fully relativistic multiconfiguration Dirac-Fock (MCDF) wavefunctions in the active space approximation, the energy levels, lifetimes, radiative rates, and oscillator strengths are determined for transitions in Mn-like W L ion. For the calculations, Breit interaction (BI) and quantum electrodynamics (QED) corrections to the levels are estimated in extensive relativistic configuration interaction (RCI) procedure, and results are reported for all electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions among the lowest 121 levels, belonging to the 3s23p63d7, 3s23p53d8, 3s3 p63d8, 3s23p43d9, and 3s3 p53d9 configurations. Comparisons are made with similar data obtained with FAC (Flexible Atomic Code) to assess the accuracy of the results. The present results are also assessed by comparison with the data compiled by the Atomic Spectra Database (ASD) of National Institute of Standards and Technology (NIST) and other existing theoretical/experimental ones for few levels in literature showing an overall good agreement. i.e., the energies agree within 0.5% with the experimental results, and the M1 transition rates agree within 5% with other theoretical results. Beside the ground configuration, almost all atomic parameters presented in this work are calculated for the first time and thus significantly increases the amount of accurate data for W L ion. Our results should be beneficial in future for identification of spectral lines, in plasma modelling, and in fusion plasma research.

Relativistic calculations on the x-ray satellites of platinum with multiply ionized L shell

The energies, electric dipole rates and oscillator strengths of Kα x-ray satellites from 1s–2p transitions in platinum with a single vacancy in the K shell and variously ionized L shell are calculated using multi-configuration Dirac Fock wavefunctions with the inclusion of Breit interaction and quantum electrodynamics effects. The computations have been performed for low ionized (Pt1+–Pt6+) and also for highly ionized (Pt69+–Pt77+) Pt. As Kα x-ray satellites with spectator vacancies only in the L shell, to the best of our knowledge, have been studied for the first time, to check the accuracy of the present work, calculations have also been performed for H-, He-, Li-, Be-, B- and F-like Pt for which theoretical data for comparison exist. The purpose of choosing a high Z element is that the detailed decay properties of heavy ions due to systematic 1s–2p transitions with spectator vacancies in the L shell using fully relativistic treatment are unexplored and the increasing need for reliable data from such ions in various applications that are under investigations in recent years. Also, for heavy ions like Pt, the contribution from the competing non-radiative transitions is less and hence the atomic properties are largely due to radiative transitions. The K X ray satellite energies, rates and resonance photo- ionization cross sections from H-, He-, Li-, Be-, B- and F-like Pt are compared with available data.

Calculations with spectroscopic accuracy: Energies and transition parameters for lines of fusion interest in Mo XXXIX

We have performed high-accuracy calculations of energy levels, wavelengths, lifetimes, line strengths and radiative rates among the 250 lowest levels of the (1s2)2s2, (1s2)2s2p, (1s2)2p2, (1s2)2snl (n=3,4,5,6; l=0,1,2,3,4,5), and (1s2)2pnl (n=3,4,5,6; l=0,1,2,3,4,5) configurations of Mo XXXIX. The second-order relativistic many-body perturbation theory (MBPT) is adopted and data are provided for all electric/magnetic-dipole (E1/M1) and electric/magnetic-quadrupole (E2/M2) transitions. Similar data for the 98 lowest levels of the n ≤ 4 configurations are also determined using fully relativistic multi-configuration Dirac-Fock (MCDF) wave functions in the active space approximation with the inclusion of a finite nuclear size, the Breit interaction, self-energy, and vacuum polarization to assess the accuracy of the calculations. Based on direct comparisons between the results obtained from these two independent methods, MBPT and MCDF, as well as with the other available theoretical values and measurements, our energies are assessed to be accurate to better than 0.02%. The accuracy for strong radiative rates is estimated to be about 5%. For Mo XXXIX, only a few transition lines have been established experimentally. The present new results considerably expand the existing data sets for this ion, and should be beneficial in analysis of experimental spectra and useful for spectroscopic diagnostics and modeling of hot plasmas, in particular in fusion devices.

Electron-impact excitation of Xe+ and polarization of its subsequent emissions

Fully relativistic distorted wave calculations have been performed to study the electron impact excitation of singly ionized xenon from its ground 5p5 (J = 3/2) state to different fine structure excited states of the 5p46s, 5p46p, 5p47s, 5p47p, 5p45d and 5p46d configurations. The ground and excited bound states of Xe+ion have been represented through the relativistic multi-configuration Dirac-Fock wave functions while projectile electron distorted continuum wave functions are obtained by solving the Dirac equations. Results are reported for the excitation cross sections and the corresponding rate-coefficients of the considered different transitions. For each transition, analytic fitting to the calculated cross sections are also provided for plasma modeling purposes. In addition, using the density matrix theory and the calculated different magnetic sub-level excitation cross sections the linear polarization of the radiation emitted by the different optically allowed transitions are obtained and presented.

Fully relativistic atomic structure calculations for W XLIV for determination of plasma diagnostic terms

We report accurate calculations of W XLIV through application of multi-configuration Dirac–Fock wave functions. We have calculated the energies for the lowest 100 fine structure levels, transition wavelengths, radiative rates, oscillator strengths, and line strengths for electric (E1) and magnetic dipole (M1) transitions with the extended average level multi-configurational Dirac–Fock method in the General-Purpose Relativistic Atomic Structure Package (GRASP). We have taken into account the electron correlations, quantum electrodynamics (QED) and Breit corrections in our calculations. We have also performed parallel calculations with the flexible atomic code (FAC) to assess the accuracy of our calculations. This is a fully relativistic code that provides a variety of atomic parameters, and (generally) yields results for energy levels and radiative rates comparable to GRASP. Our calculated results match well with experimentally observed results that are obtained in ASDEX upgrade Tokamak. Additionally, we have also provided the line intensity ratios and electron density for W XLIV, which is useful and important in plasma diagnostics and modeling in future International Thermonuclear Experimental Reactor (ITER) experiments. We believe that our results would be beneficial in the areas of fusion plasma research and astrophysical investigations and applications.

QED effects in 1s and 2s single and double ionization potentials of the noble gases

We present calculations on the quantum electrodynamics (QED) effects in 1s and 2s single and double ionization potentials of noble gases from Ne to Rn as perturbations on relativistic four-component Dirac-Fock wavefunctions. The most dominant effect originates from the self-energy of the core-electron that yields corrections of similar order as the transverse interaction. For 1s ionization potentials, a match within few eV against the known experimental values is obtained, and our work reveals considerable QED effects in the photoelectron binding energies across the periodic table-most strikingly even for Ne. We perform power-law fits for the corrections as a function of Z and interpolate the QED correction of ∼-0.55 eV for S1s. Due to this, the K-edge electron spectra of the third row and below need QED for a match in the absolute energy when using state-of-the-art instrumentation.

Kα1,2x-ray linewidths, asymmetry indices, and[KM]shake probabilities in elements Ca to Ge and comparison with theory for Ca, Ti, and Ge

We have investigated systematically the $K\ensuremath{\alpha}$ x-ray spectra in elements from Ca to Ge within Berger's two-Lorentzian functions model, using a high-resolution antiparallel double-crystal x-ray spectrometer, in order to obtain in detail the physical meaning of the asymmetry in the spectral profiles. The overall tendency of the corrected full width at half maximum of the $K{\ensuremath{\alpha}}_{1}$ and $K{\ensuremath{\alpha}}_{2}$ lines as a function of $Z$, as well as the linewidths, are in good agreement with the data reported in the literature. It is found, from both the experiments and calculation, that satellite lines arising from shake-off appear between the $K{\ensuremath{\alpha}}_{1}$ and $K{\ensuremath{\alpha}}_{2}$ lines. The asymmetry index of $K{\ensuremath{\alpha}}_{1}$ in $3d$ elements from Sc to Zn is ascribed to the existence of a $3d$ spectator hole. Moreover, the observed Sc $K{\ensuremath{\alpha}}_{1}$ line shows a symmetric profile unlike the Ti $K{\ensuremath{\alpha}}_{1}$ line profile, this discrepancy being explained by the existence of satellite lines on both sides of the Sc profile. Our experimental results yielded around 25% probability for the probability of shake processes creating a second hole in the $3p$ or $3d$ subshells in Sc and around 18% probability for creating a second hole the in the $3p$ in Ca. For the latter process our calculated value, using multiconfiguration Dirac-Fock wave functions and the sudden approximation yielded a 10% probability.

Revised and extended calculations of level energies, M1 and E2 radiative rates for highly charged tungsten ions from W57+ to W60+

We have applied systematically enlarged multiconfiguration Dirac–Fock wavefunctions using Grasp2K to calculate the transition energies, oscillator strengths and transition probabilities for fine structure M1 and E2 transitions between the low-lying levels of the 3s23p5, 3s23p4, 3s23p3 and 3s23p2 configurations of highly charged tungsten ions from to . Large wavefunction expansions are applied to calculate the transition probabilities, which are indispensable for calculating various plasma parameters accurately. In the present calculations, our theoretical data agrees well with that obtained in precise electron beam ion trap measurements, and is therefore important for the identification of weak forbidden lines for plasma diagnostic applications.